# cancellative semigroup

Let $S$ be a semigroup.

$S$ is *left cancellative* if, for all $a,b,c\in S$, $ab=ac\Rightarrow b=c$

$S$ is *right cancellative* if, for all $a,b,c\in S$, $ba=ca\Rightarrow b=c$

$S$ is *cancellative* if it is both left and right cancellative.

## 1 Relationship to some other types of semigroup

This is a generalisation of groups, and in fact being cancellative is a necessary condition for a semigroup to be embeddable in a group.

Note that a non-empty semigroup is a group if and only if it is cancellative and regular^{}.

$S$ is *weakly cancellative* if, for all $a,b,c\in S$, $(ab=ac\&ba=ca)\Rightarrow b=c$

A semigroup is completely simple if and only if it is weakly cancellative and regular.

## 2 Individual elements

An element $x\in S$ is called *left cancellative* if, for all $b,c\in S$, $xb=xc\Rightarrow b=c$

An element $x\in S$ is called *right cancellative* if, for all $b,c\in S$, $bx=cx\Rightarrow b=c$

Title | cancellative semigroup |

Canonical name | CancellativeSemigroup |

Date of creation | 2013-03-22 14:25:09 |

Last modified on | 2013-03-22 14:25:09 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 9 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20M10 |

Synonym | cancellation semigroup |

Related topic | CancellationIdeal |

Defines | cancellative |

Defines | weakly cancellative |

Defines | left cancellative |

Defines | right cancellative |

Defines | weakly cancellative semigroup |

Defines | left cancellative semigroup |

Defines | right cancellative semigroup |